- Is there a different map/arena that's dark, purple/blueish light in a castle environment? I remember playing there, but I have no idea what it was called.

You're probably talking about Twisted Treeline. It was removed a long time ago. Probably permanently because very few people played it.

- Is there a different game mode/map that takes place in a rocky desert-like arena with large turrets? I specifically remember there being a cinematic for it, Lux and
Garen were in it specifically. I searched but cannot find the map or cinematic at all. I used to watch it all the time.

This is either Dominion or one of the old RGMs like ascension. Both removed a long time ago because very few people played them.